Foundations of mathematics - their past, present and future.
The 2014 Paul Bernays Lectures at ETH Zurich. Sep. 9-10, 2014.
I. To the history of the conception. SlidesVideo
II. The story of set theory (so far). SlidesVideo
III. Univalent Foundations. SlidesVideo
Note: there is a lot in the videos that is not in the slides.

Computer proof assistants - the future of mathematics.
Lecture at the NUS, Singapore. August 27, 2014.Slides

How I became interested in foundations of mathematics.
Lecture at the 8th Asian Science Camp in Singapore. August 25, 2014.Slides
Note: It was a lecture for undergraduates. Here is one of the posters that the listeners made after the lecture.

Dependently-typed systems and their models.
Lecture at a workshop in IHP, Paris. June 6, 2014. Slides​
Note: this is a very messy workshop talk.

Univalent Foundations - new type theoretic foundations of mathematics.
Lecture at IHP, Paris. April 22, 2014.2014_04_22_slides.pdfVideo on YouTude
Note: this is a research talk not a systematic presentation of a subject.

Univalent Foundations.
Lecture at IAS, Princeton. Mar. 26, 2014.2014_IAS.pdf
Note: This is a general audience talk which is partly autobiographical
with the discussion of the role of mistakes in mathematics. Here is an article that was made from it.

A type system with two kinds of identity types,
Talk at Andre Joyal's 70th birthday conference (IAS), Feb. 25, 2013HTS_slides.pdfHTS_notes.pdf
Note: This is the talk where the idea of HTS was introduced.

Two lectures
Wuhan University 110th anniversary, November-December 2003
What is most important for mathematics in the near future? (Notes)
About mathematics and motivic homotopy theory (Transparencies)